Cross-over trials

Topics to be covered

What you will learn
- The paired t-test
- The AB/BA cross-over
- Period effects and carry-over
- When should you use or not use a cross-over
- Cross-over trials with a binary outcome
- Cross-over trials with more than two treatments

Speaker notes

Cross-over trials provide a very powerful approach for comparing two treatment conditions. Research subjects get both treatment conditions, which we will label arbitrarily as A and B. A randomly selected half get A followed by B and the other half gets B followed by A. This is generally referred to as the AB/BA design. Because each subject serves as their own control, you remove a large amount of variation. Often a cross-over trial can produce good results, even with a few dozen patients.

Cross-over trials require that the condition being studied is chronic in nature, and that the effect of the first treatment given does not carry over into the second treatment evaluation. There are important ethical and practical concerns about designing cross-over trials.

In this talk, you will see a review of the paired t-test, followed by an analysis of the AB/BA crossover design as a paired t-test. Then you will see how to account for the order in which the treatments were given.

After discussing the classical AB/BA cross-over design, you will see how to extend the cross-over trial to a binary outcome measure and to more than two treatments.

Feedback

Mixed effects model
Heading for 14 and 15
Graphs 34 and earlier
Lobotomy
Don’t show code
Simplify the R output
No need for SAS
Kim Love did classical experimental design webinar
Have I defined a block?
Introduce the cross-over using a diagram
Why do we need to do both orders?
Blog post using long and wide format, https://www.theanalysisfactor.com/wide-and-long-data/
Fix tibble on slide 31
Fix font sizes on plots 34 and others
Put higher is better on every plot
Define double blind, won’t know until they get out of the plane
Define washout period.
State versus trait, trait is immutable
Slide 40, what if adding treatments is the point. Physical therapy plus
Introduce the concept of a parallel groups design
Slide 41 what if
Slide 46 maybe use some type of random effects logistic regression.
Slide 50 CBAD in second row
Why are there no interactions in slide on page 35.

Quotations from two great philosophers

“A man cannot step into the same river twice, because it is not the same river, and he is not same man.” - Heraclitus, 544 B.C.
“You must unlearn what you have learned.” Yoda, 1980 A.D.

Paired t-test data

Descriptive statistics on wife’s age

variable	mean	sd	n
wAge	40.7	11.4	170

Descriptive statistics on husband’s age

variable	mean	sd	n
hAge	42.6	11.6	199

Descriptive statistics on age gap

variable	mean	sd	n
age_gap	2.2	4.1	170

Paired t-test on couples ages

Difference mean (\(\bar{D}\)) = 2.2
Difference standard deviation (\(S_D\)) = 4.1
Difference standard error (\(S_D / \sqrt{n}\)) = 0.3
Test statistic \(\Big(\frac{\bar{D}}{S_D/\sqrt{n}}\Big)\) = 7.15
p<0.001

Analyze as completely randomized block design

Analysis of Variance Table

Response: age
           Df Sum Sq Mean Sq F value    Pr(>F)    
spouse      1    425  424.71  51.148 2.474e-11 ***
pair      169  43974  260.20  31.337 < 2.2e-16 ***
Residuals 169   1403    8.30                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Restructured data for a completely randomized block design

Mixed model/random coefficient analysis

Uses long format
Dependent variable: age
Fixed effect: spouse
Random intercept: pair

Break #1

What you have learned
- The paired t-test
What’s coming next
- The AB/BA cross-over

Parallel groups versus the AB/BA cross-over

https://statistically-funny.blogspot.com/2021/03/in-clinical-trials-you-can-have-it-both.html

Data layout

Alternate data layout

Descriptive statistics, period 1

variable	mean¹	sd	n
period1	312.3	82.8	13
¹ Larger values imply better outcomes

Descriptive statistics, period 2

variable	mean¹	sd	n
period2	324.6	67.9	13
¹ Larger values imply better outcomes

Descriptive statistics, period gap

variable	mean¹	sd	n
period_gap	12.3	61.0	13
¹ Defined as period2 - period1

Descriptive statistics, formoterol

variable	mean¹	sd	n
formoterol	341.2	59.7	13
¹ Larger values imply better outcomes

Descriptive statistics, salbutamol

variable	mean¹	sd	n
salbutamol	295.8	82.9	13
¹ Larger values imply better outcomes

Descriptive statistics, drug gap

variable	mean¹	sd	n
drug_gap	45.4	40.6	13
¹ Defined as formoterol - salbutamol

Paired t-test for the AB/BA cross-over data

Difference mean (\(\bar{D}\)) = 45.4
Difference standard deviation (\(S_D\)) = 40.6
Difference standard error (\(S_D/\sqrt{n}\)) = 11.3
Test statistic \(\Big(\frac{\bar{D}}{S_D/\sqrt{n}}\Big)\) = 4.03
p<0.001

Analysis using a completely randomized block design

Analysis of Variance Table

Response: pef
          Df Sum Sq Mean Sq F value    Pr(>F)    
treatment  1  13388 13388.5  16.250  0.001666 ** 
id        12 115213  9601.1  11.654 7.935e-05 ***
Residuals 12   9887   823.9                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Break #2

What you have learned
- The AB/BA cross-over
What’s coming next
- Period effects and carry-over

Accounting for period effects

Improves precision
Adjusts for imbalances
Requires layout used for completely randomized block design

Why might you see period effects

Temporal trends
Seasonal effects
Carry-over: Persistence of a treatment applied in one period in a subsequent period of treatment
- Physical persistence
- Curative effect

Layout for asthma data

Counts for layout data

treatment	order	period	n
formoterol	for/sal	1	7
formoterol	sal/for	2	6
salbutamol	for/sal	2	7
salbutamol	sal/for	1	6

Means, 1

# A tibble: 2 × 3
  order   period1_mean period2_mean
  <chr>          <dbl>        <dbl>
1 for/sal         337.         306.
2 sal/for         283.         346.

Interaction, 1

What this graph would look like without a period effect

Means, 2

# A tibble: 2 × 3
  order   formoterol_mean salbutamol_mean
  <chr>             <dbl>           <dbl>
1 for/sal            337.            306.
2 sal/for            346.            283.

Interaction, 2

Accounting for period effects

Analysis of Variance Table

Response: pef
          Df Sum Sq Mean Sq F value    Pr(>F)    
order      1    335   335.2  0.4467  0.517697    
id        11 114878 10443.5 13.9171 6.495e-05 ***
period     1    985   984.6  1.3121  0.276323    
treatment  1  14036 14035.9 18.7044  0.001205 ** 
Residuals 11   8254   750.4                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Break #3

What you have learned
- Period effects and carry-over
What’s coming next
- When should you use or not use a cross-over

Systematic overview of parachute use

When you cannot use a cross-over trial

Acute conditions
Mortality/morbidity
Training/education
Irreversible treatments
Persistent effects

When you can use a cross-over trial

Chronic conditions
Short-term benefits
Adequate wash-out period
- Drugs with a short half-life

Ethical concerns

Can you ask a patient to endure two treatments?
Can you leave the patient untreated during the wash-out period

What is you have a carry-over effect?

No easy answer
- Adjustments have poor power
- Between subject comparison
Unclear where to make adjustments
- Two active treatments:
  - Split the difference
- Active treatment versus placebo:
  - Assign the carry-over to placebo
- Carry-over could an interaction instead

Change test if carry-over is detected

Test hypothesis of no carry-over
- If accept null, run the cross-over
- If reject null, look at first period only
Not recommended
- Limited power to test carry-over
- First period analysis loses to much power/precision

Break #4

What you have learned
- When should you use or not use a cross-over
What’s coming next
- Cross-over trials with a binary outcome

Example of a binary outcome

Descriptive statistics

	F-	F+
S-	2	15
S+	1	6

Analysis of binary cross-over data

Ignoring period effects
- McNemar’s test
Incorporating period effects
- Mainland-Gart test (hard to find)

Example of McNemar’s test

Break #5

What you have learned
- Cross-over trials with a binary outcome
What’s coming next
- Cross-over trials with more than two treatments

Three treatment cross-over trial

Show example data from page 124 of Senn

Possible sequences

Three treatments
- ABC, ACB, BAC, BCA, CAB, CBA
Four treatments
- ABCD, ACBD, BACD, BCAD, CABD, CBAD,
- ABDC, ACDB, BADC, BCDA, CADB, CBAD,
- ADBC, ADCB, BDAC, BDCA, CDAB, CDBA,
- DABC, DACB, DBAC, DBCA, DCAB, DCBA

Latin square design to reduce the number of sequences

Three treatments
- Sequence 1: A B C
- Sequence 2: B C A
- Sequence 3: C A B
Four treatments
- Sequence 1: A B C D
- Sequence 2: B C D A
- Sequence 3: C D A B
- Sequence 4: D A B C

The definitive guide

Nice reference

Summary

What you have learned
- The paired t-test
- The AB/BA cross-over
- Period effects and carry-over
- When should you use or not use a cross-over
- Cross-over trials with a binary outcome
- Cross-over trials with more than two treatments